finding poles for a complex rational function

Question

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the residue thm or Cauchy integral formula. However, I think I might be losing it since I'm having trouble finding the roots. Can anyone give me a step by step for factoring or otherwise?

Answer 1

Hint: See when $$z^4+6z^2+1 = (z^2+3)^2 - 8 = 0$$

Answer 2

Consider the quadratic in $z^2$ and using your favorite method for solving we have $z^2 = -3 \pm 2\sqrt{2}$. Now you can write down the four values for $z$. The partial fractions are a bit messy.